
For \(\gamma <0\) ,
by lemma 1.2.9 in [1]}
\(\frac{{a}_N}{\alpha _N} & \rightarrow -\frac{1}{\gamma },\\\frac{\beta _N-{b}_N}{{a}_N} & = \dfrac{U_F(N)- \theta _F}{\theta _F - U_F(N)} = -1.\) 

For \(\gamma >0\) 
by lemma 1.2.9 in [1]}
\( \dfrac{a_N}{\alpha _N} & \rightarrow \dfrac{1}{\gamma },\\\dfrac{\beta _N- b_N}{a_N} & = \dfrac{U_F(N)-0}{U_F(N)} =1.\) 

For \(\gamma =0\)
\(a_N & = \alpha _N, \\b_N & = \beta _N .\) 

